Mathematics
Differential Equations
Publisher: Brooks Cole, 2005, 848pp, 3rd ed.
Incorporating a modeling approach throughout, this exciting introductory textbook emphasizes concepts and shows that the study of differential equations is a beautiful application of the ideas and techniques of calculus to everyday life. Students will generally attack a given equation from three different points of view to obtain an understanding of the solutions: qualitative, numeric, and analytic.
Table of contents
| 1 | First-Order Differential Equations | 1 |
| 1.1 | Modeling via Differential Equations | 2 |
| 1.2 | Analytic Technique: Separation of Variables | 20 |
| 1.3 | Qualitative Technique: Slope Fields | 36 |
| 1.4 | Numerical Technique: Euler's Method | 53 |
| 1.5 | Existence and Uniqueness of Solutions | 65 |
| 1.6 | Equilibria and the Phase Line | 76 |
| 1.7 | Bifurcations | 96 |
| 1.8 | Linear Differential Equations | 113 |
| 1.9 | Changing Variables | 123 |
| Labs for Chapter 1 | 138 | |
| 2 | First-Order Systems | 147 |
| 2.1 | Modeling via Systems | 148 |
| 2.2 | The Geometry of Systems | 165 |
| 2.3 | Analytic Methods for Special Systems | 183 |
| 2.4 | Euler's Method for Systems | 194 |
| 2.5 | The Lorenz Equations | 209 |
| Labs for Chapter 2 | 216 | |
| 3 | Linear Systems | 225 |
| 3.1 | Properties of Linear Systems and the Linearity Principle | 226 |
| 3.2 | Straight-Line Solutions | 250 |
| 3.3 | Phase Planes for Linear Systems with Real Eigenvalues | 266 |
| 3.4 | Complex Eigenvalues | 282 |
| 3.5 | Special Cases: Repeated and Zero Eigenvalues | 301 |
| 3.6 | Second-Order Linear Equations | 316 |
| 3.7 | The Trace-Determinant Plane | 333 |
| 3.8 | Linear Systems in Three Dimensions | 346 |
| Labs for Chapter 3 | 362 | |
| 4 | Forcing and Resonance | 369 |
| 4.1 | Forced Harmonic Oscillators | 370 |
| 4.2 | Sinusoidal Forcing | 385 |
| 4.3 | Undamped Forcing and Resonance | 397 |
| 4.4 | Amplitude and Phase of the Steady State | 409 |
| 4.5 | The Tacoma Narrows Bridge | 421 |
| Labs for Chapter 4 | 431 | |
| 5 | Nonlinear Systems | 437 |
| 5.1 | Equilibrium Point Analysis | 438 |
| 5.2 | Qualitative Analysis | 457 |
| 5.3 | Hamiltonian Systems | 470 |
| 5.4 | Dissipative Systems | 488 |
| 5.5 | Nonlinear Systems in Three Dimensions | 510 |
| 5.6 | Periodic Forcing of Nonlinear Systems and Chaos | 518 |
| Labs for Chapter 5 | 535 | |
| 6 | Laplace Transforms | 541 |
| 6.1 | Laplace Transforms | 542 |
| 6.2 | Discontinuous Functions | 554 |
| 6.3 | Second-Order Equations | 563 |
| 6.4 | Delta Functions and Impulse Forcing | 577 |
| 6.5 | Convolutions | 585 |
| 6.6 | The Qualitative Theory of Laplace Transforms | 594 |
| Labs for Chapter 6 | 603 | |
| 7 | Numerical Methods | 607 |
| 7.1 | Numerical Error in Euler's Method | 608 |
| 7.2 | Improving Euler's Method | 621 |
| 7.3 | The Runge-Kutta Method | 629 |
| 7.4 | The Effects of Finite Arithmetic | 640 |
| Labs for Chapter 7 | 644 | |
| 8 | Discrete Dynamical Systems | 647 |
| 8.1 | The Discrete Logistic Equation | 648 |
| 8.2 | Fixed Points and Periodic Points | 661 |
| 8.3 | Bifurcations | 670 |
| 8.4 | Chaos | 679 |
| 8.5 | Chaos in the Lorenz System | 687 |
| Labs for Chapter 8 | 693 | |
| Appendices | 699 | |
| A | First-Order Linear Equations Revisited | 700 |
| B | Complex Numbers and Euler's Formula | 711 |
| Hints and Answers | 717 | |
| Index | 777 |